1,2Department of Computer Science and Engineering, K.S.Rangasamy College of Technology,Tiruchengode-637215, India*E-mail: nallsam@rediffmail.com

3D-Link India Ltd, Bangalore, India4Department of Production Engineering, National Institute of Technology Tiruchirappalli, IndiaABSTRACTThis paper deals with generating of an optimized route for multiple Vehicle routing Problems(mVRP). We used a methodology of clustering the given cities depending upon the number of vehicles and eachcluster is allotted to a vehicle. k- Means clustering algorithm has been used for easy clustering of the cities. Inthis way the mVRP has been converted into VRP which is simple in computation compared to mVRP. Afterclustering, an optimized route is generated for each vehicle in its allotted cluster. Once the clustering had beendone and after the cities were allocated to the various vehicles, each cluster/tour was taken as an individualVehicle Routing problem and the steps of Genetic Algorithm were applied to the cluster and iterated to obtainthe most optimal value of the distance after convergence takes place. After the application of the variousheuristic techniques, it was found that the Genetic algorithm gave a better result and a more optimal tour formVRPs in short computational time than other Algorithms due to the extensive search and constructive natureof the algorithm.Keywords: Multiple vehicle routing problem, k-means clustering, genetic algorithm, and combinatorialoptimization.

1. INTRODUCTIONProblems of combinatorial optimization are characterized by their well-structured problem definition aswell as by their huge number of action alternatives in practical application areas of reasonable size [9]. Utilizingclassical methods of Operations Research often fails due to the exponentially growing computational effort.Therefore, in practice heuristics and meta-heuristics are commonly used even if they are unable to guarantee anoptimal solution.1.1 Scope and objectives of researchA careful analysis of literature on the variants and methodologies of combinatorial optimizationproblems reveals that some of the variants of combinatorial optimization problems are yet to be explored tosolve using meta-heuristics techniques [9],[10]. These include:• Multiple Vehicle Routing Problem (mVRP) and multiple Traveling Salesman Problem (mVRP)• mVRP with balanced allocation of nodes with single objective or multiple objectivesMany of the authors

[2] have suggested the use of a constructive heuristic to obtain good initialsolutions for a meta-heuristic so that its convergence can be accelerated. Only a few authors have considered theuse of hybrid approaches to solve different variants of combinatorial optimization problems.1.2 Vehicle routing problemIn VRP a number of cities have to be visited by a vehicle which must return to the same city where itstarted. In solving the problem one tries to construct the route so that the total distance traveled is minimized.Every vehicle starts from the same city, called depot and must return at the end of its journey to this city again.If n is the number of cities to be visited then (n-1)! is the total number of possible routes. As theamount of input data increases the problem increases in complexity, thus the computational time needed rendersthis method impractical for all but a smaller number of cities. Rather than considering all possible tours,heuristic algorithms for solving the VRP are capable of substantially reducing the number of tours to be takeninto consideration.ISSN: 0975-5462R. Nallusamy et al. /International Journal of Engineering Science and TechnologyVol.1(3), 2009, 129-1351301.3 Multiple vehicle routing problemA generalization of the well-known vehicle routing Problem is the multiple vehicle routing problem,which consists of determining a set of routes for m vehicles. The mVRP can in general be defined as follows:Given a set of nodes, let there be m vehicle located at a single depot node. The remaining nodes (cities) that areto be visited are called intermediate nodes. Then, the mVRP consists of finding tours for all m vehicles, whichall start and end at the depot, such that each intermediate node is visited exactly once and the total cost ofvisiting all nodes is minimized.2. REVIEW

OF

EXISTING

WORKMany methods have been suggested for obtaining optimized route[2,3]. Rizzoli et al.[1] have focusedon the Application of Ant Colony Optimization on the Vehicle Routing Problem and its real world application.Potvin[2] has worked on the survey of the genetic algorithms in his study he has given simple geneticalgorithms and various extensions for solving Traveling Salesman Problem (TSP). He has worked both on therandom and the classical problems [6].Schabauer, Schikuta, and Weishaupl [3] have worked on to solve traveling salesman problemheuristically by the parallelization of self-organizing maps on cluster architectures. Allan Larsen hasworked on the dynamic factors of vehicle routing problem. He has investigated the dynamics of the vehiclerouting problem in order to improve the performances of existing algorithms and as well as developed newalgorithms [4]. Jorg Homberger and Hermann Gehring have worked on vehicle routing problems on timewindows. In this they have designed an optimal set of routes that will service the entire customer with constrainsbeing taken care of properly. Their objective function minimizes both the total distance traveled and the numberof salesmen being used[5].Al-Dulaimi and Ali [6] have proposed a software system to determine the optimal route of the travelingsalesman using Genetic Algorithm (GA). The system proposed starts from a matrix of the calculated Euclideandistances to the cities to be visited by the salesman. The new generations are formed from this until proper pathis obtained. Chao, Ye and Miao [7] have developed a two level genetic algorithm which favors neither intra-cluster path or inter-cluster path. The results from the study indicate that the algorithm proposed is moreeffective than the existing algorithms.A.E. Carter, C.T. Ragsdale have developed a new approach to solve mTSP. The method proposes newset of chromosomes and related operators for the mTSP and compares theoretical properties and computationalperformance of the proposed technique. The computational technique shows that the newer technique results inthe smaller search space and produces better solutions [8].Mitrovic-Minic and Krishnamurti[7] have worked onto to find the lower and upper bound required forthe number of vehicles to serve all locations for multiple traveling Salesman problem with time windows. Theyhave introduced two types of precedence graphs namely the start time precedence graphs and the end timeprecedence graphs. The bounds are generated by covering the precedence graphs with minimum number ofpaths. The bounds which are tight and loose are compared and the closeness of such instances were discussed.Researchers on the VRP have proved that the VRP is a NP-complete combinatorial optimizationproblem. They have theorized that if an algorithm is guaranteed to find the optimal solution in a polynomialtime for the VRP, then efficient algorithms could also be found for all the other NP-complete problems[10]-[15].1.4 Research gap and proposed workFrom the review, we understood that most of the problems involved solving the conventional vehiclerouting problem or traveling salesman problem using exact as well as meta-heuristic methods for solving thesame. They however scarcely dealt with the multiple vehicle routing Problem which represents the realistic caseof more than one vehicle. To the best of our knowledge, from the literature review, no efficient meta-heuristicalgorithms exist for the solution of large-scale mVRPs. Also, the solution procedures based on transforming themVRP to the standard VRP do not seem efficient, since the resulting VRP is highly degenerate, especially withthe increasing number of vehicles. Hence, an analysis is made and an heuristic is formed to transform mVRP toVRP and to optimize the tour of an individual. We decided to deal with the less frequently approached and morerealistic multiple vehicle routing problem along with a specialized clustering heuristic, namely k-meansclustering algorithm3. PROBLEM

BACKGROUND

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PROBLEM

FORMULATIONThe mathematical structure of the VRP is a graph where the cities are the nodes of the graph.Connections between pairs of cities are called edges and each edge has a cost associated with it which can beISSN: 0975-5462R. Nallusamy et al. /International Journal of Engineering Science and TechnologyVol.1(3), 2009, 129-135131distance, time or other attribute. If n is the input number of vertices representing cities, for a weighted graph G,the VRP problem is to find the cycle of minimum costs that visit each of the vertices of G exactly once.There are many mathematical formulations for the VRP, employing a variety of constraints that enforcethe requirements of the problem. The following notation is used: n-The number of cities to be visited; thenumber of nodes in the network; i, j, k- Indices of cities that can take integer values from 1 to n; t-The timeperiod, or step in the route between the cities; xijt-1 if the edge of the network from i to j is used in step t of theroute and 0 otherwise; dij-The distance or cost from city i to city j. The following is an example of one linearprogramming formulations of the VRP problem:The objective function (Z) is to minimize the sum of all costs (distances) of all of the selected elementsof the tour:n n nij ijti 1 j i t 1Z d x= = ==∑∑∑(1)The tour is subject to the following constraints. For all values of t, exactly one arc must be traversed,hence:ijti jx 1 for all t=∑∑(2)For all cities, there is just one other city which is being reached from it, at some time, hence:ijtj tx 1 for all i=∑∑(3)For all cities, there is some other city from which it is being reached, at some time, henceijti tx 1 for all j=∑∑(4)When a city is reached at time t, it must be left at time t+1, in order to exclude disconnected sub-toursthat would otherwise meet all of the above constraints. These sub-tour elimination constraints are formulated as:

ijt jkt 1i kx x for all j and t+=∑ ∑(5)In addition to the above constraints the decision variables are constrained to be integer values in therange of 0-1:ijt0 x 1≤ ≤(6)4. MATERIALS

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METHODS4.1. AssumptionsAll the salespersons have to start from a common depot and after traveling through a set of cities, theyshould return back to the starting depot. There are no capacity constraints and no cost constraints. But, all thecities must be visited by any one of the salesperson and each salesperson has to visit a particular city exactlyonce.4.2. Transformation of mVRP to VRPThe search space for the solution increase as the number of cities decreases and vice-versa. If there areN cities then the search space will be N! and the computational time also high accordingly. Hence to reduce theburden of mathematical complexity N value should be reduced and this is achieved by clustering. The followingheuristics were used for solving the given 180 cities 6 vehicles problem. City number 100 is considered to be theheadquarters of all the vehicles.4.3. k-means clusteringSimply speaking k-means clustering is an algorithm to classify or to group the objects based onattributes/features into k number of group. k is a positive integer number. The grouping is done by minimizingthe sum of squares of distances between data and the corresponding cluster centroid [2].The main idea is to define k centroids, one for each cluster. These centroids should be placed in acunning way because of different location causes different result. So, the better choice is to place them as muchISSN: 0975-5462R. Nallusamy et al. /International Journal of Engineering Science and TechnologyVol.1(3), 2009, 129-135132as possible far away from each other. The next step is to take each point belonging to a given data set andassociate it to the nearest centroid.When no point is pending, the first step is completed and an early groupage is done. At this point weneed to re-calculate k new centroids as barycenters of the clusters resulting from the previous step. After wehave these k new centroids, a new binding has to be done between the same data set points and the nearest newcentroid. A loop has been generated. As a result of this loop we may notice that the k centroids change theirlocation step by step until no more changes are done. In other words centroids do not move any more.This produces a separation of the objects into groups from which the metric to be minimized can becalculated. Although it can be proved that the procedure will always terminate, the k-means algorithm does notnecessarily find the most optimal configuration, corresponding to the global objective function minimum. Thealgorithm is also significantly sensitive to the initial randomly selected cluster centers. The k-means algorithmcan be run multiple times to reduce this effect.The algorithm is composed of the following steps:• Place k points into the space represented by the objects that are being clustered. These pointsrepresent initial group centroids• Assign each object to the group that has the closest centroid• When all objects have been assigned, recalculate the positions of the K centroids• Repeat steps 2 and 3 until the centroids no longer move. This produces a separation of theobjects into groups from which the metric to be minimized can be calculated4.4. Application of GA to the given mVRPGenetic algorithms emulate the mechanics of natural selection by a process of randomized dataexchange. The fact that they are able to search in a randomized, yet directed manner, allows them to reproducesome of the innovative capabilities of natural systems. GAs work by generating a population of numeric vectorscalled chromosomes, each representing a possible solution to a problem. The individual components within achromosome are called genes. New chromosomes are created by crossover or mutation. Chromosomes are thenevaluated according to a fitness function, with the fittest surviving and the less fit being eliminated. The result isa gene pool that evolves over time to produce better and better solutions to a problem. The GAs search processtypically continues until a pre-specified fitness value is reached, a set amount of computing time passes or untilno significant improvement occurs in the population for a given number of iterations. The key to find a goodsolution using a GA lies in developing a good chromosome representation of solutions to the problem.4.5. Algorithm for genetic algorithmP = Generate Initial Population of Solutions;While (stopping criterion not met)For (X ∈ P) C(X) = Evaluate Cost of X;P′ = Select Fittest Individuals from P to Form Mating Pool;P′′ = ∅;Repeat (until enough children produced)Select X1 and X2 at Random From P′;Apply Mating Procedures to X1 and X2 to Produce Xchild;P′′ = P′′ ∪ Xchild;End Repeat;For (X ∈ P′′)Apply Random Mutation to X;End For;P = P′′;End While;Output C(X) where X is fittest individual in P;End.There is an optimal set of cities allocated to every vehicle after performing the k-means clusteringalgorithm. Genetic algorithm is now applied for every such cluster of cities and iteration is performed severaltimes to find an optimal value for the distance traveled by each vehicle.4.6. Formulation of the fitness valueA fitness function is a particular type of objective function that quantifies the optimality of a solutionso that that particular chromosome may be ranked against all the other chromosomes. Optimal chromosomes, orISSN: 0975-5462R. Nallusamy et al. /International Journal of Engineering Science and TechnologyVol.1(3), 2009, 129-135133at least chromosomes which are more optimal, are allowed to breed and mix their datasets by any of severaltechniques, producing a new generation that will be even better. The shorter the route, the higher is the fitnessvalue. Hence, we formulated the fitness value or function as the inverse of the reciprocal of the distance traveledin each sequence. Hence, fitness value or function f = 1/di, where diis the distance traveled by a vehicle aftercovering all the cities allocated to him.4.7. Selection of the initial populationAfter optimally assigning a definite group of cities to a vehicle using the clustering meta-heuristic, agroup of 10 sequences of all the cities in the cluster are selected from the universal population of all possiblesequences. This is performed randomly by using probabilistic selection based on the favorable fitness value.4.8. Crossover and Cross over probabilityThe population is arranged in descending order of the sequence’s fitness value. The top 7 members ofthe list are selected and random values are assigned to each chromosome using random number generationfunction. In our problem we took the crossover probability as 0.8. Simple chromosome 'crossover' is chosen forour reproduction scheme because it is the simplest. Finally, we generate an initial population of 200 randomchromosomes and run the program. We are astounded as the program produces nothing of use.From the review we identified that the order in which data is encoded in the chromosome is importantfor VRP and that a simple crossover reproduction mechanism is not suitable in these circumstances. With this inmind, we need a scheme analogous to simple crossover, but one which preserves the solution viability whileallowing the exchange of ordering information. One such scheme is Partially Matched Crossover. In thisscheme, a crossing region is chosen by selecting two crossing sites. This type of partial matched crossover isdone for every chromosome with the next best to yield an offspring. The offspring replaces the parentchromosome if the fitness value for the former is higher than that of the later.4.9. Mutation probabilityFrom the population of the chromosomes modified after crossover, a set of chromosomes are selectedfor mutation based on the mutation probability. In our problem we took the mutation probability as 0.1. Afterperforming the mutation on the selected chromosomes, the next set of 10 chromosomes is taken as the initialpopulation for the next iteration. The previous crossover and mutation steps are repeated for several iterationstill the fitness value of the best chromosome in a given population converges to a constant value. This yields theresult for the optimal distance traveled by one vehicle.5. IMPLEMENTATION,

RESULTS

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DISCUSSIONThe solution to the problem is attained using two-stage heuristics. The first-stage involves theconversion of a mVRP to VRP using k-means Clustering algorithm. Even though we get, the cities allocated to avehicle, it is important to generate a tour and improve it. Second-stage is meta-heuristic approach comprisingGA to optimize the tour for m vehicles. Then the effective solution generated by GA is studied and is comparedwith the results obtained from other methods. This gave an advantage of choosing a consistent approach toparticular types of problems. The problem was implemented in MATLAB 7.0 with Pentium IV processorsystem.Here we used randomly generated coordinates of 180 cities within 35X35 square units space. Afterperforming k-means clustering algorithm, GA is applied. Figure 1 and 2 show the convergence of results in GA.Table 1 shows the results of k-means clustering. Table 2 shows the results of GA with other algorithms.Algorithm was applied to the given problem iteratively; we found that optimal results of the distance wereobtained after performing approximately 300 iterations.Table 1. Cities allocated to the 6 vehiclesCluster /vehicle Number of Cities allocated1 442 423 254 255 76 37